Unraveling R0: Considerations for Public Health Applications

We assessed public health use of R₀, the basic reproduction number, which estimates the speed at which a disease is capable of spreading in a population. These estimates are of great public health interest, as evidenced during the 2009 influenza A (H1N1) virus pandemic.We reviewed methods commonly used to estimate R₀, examined their practical utility, and assessed how estimates of this epidemiological parameter can inform mitigation strategy decisions.In isolation, R₀ is a suboptimal gauge of infectious disease dynamics across populations; other disease parameters may provide more useful information. Nonetheless, estimation of R₀ for a particular population is useful for understanding transmission in the study population. Considered in the context of other epidemiologically important parameters, the value of R₀ may lie in better understanding an outbreak and in preparing a public health response.During the spring of 2009, the 2009 H1N1 influenza pandemic began in North America and quickly spread around the world, sparking great interest in potential mitigation strategies for the first influenza pandemic in more than 40 years. Research focused on interventions such as social distancing that could be applied before a specific monovalent H1N1 vaccine became available in the fall of 2009. During the initial wave of the 2009 H1N1 outbreak, teams of modelers from around the world gathered available data from Mexico to estimate several of the novel virus’s characteristics.1,2 Efforts focused on the rapid estimation of the basic reproduction number, or R₀, of this virus. R₀ is a theoretical parameter that provides some information regarding the speed at which a disease is capable of spreading in a specific population. First estimates were published online by early May 2009.1,2 Estimates of R₀ continue to be published from other countries and as more data become available.3–11As an indicator of the interest in publications concerning R₀, an early publication on the pandemic potential of the 2009 H1N1 strain by Fraser et al.1 has garnered 654 citations as of February 7, 2013. Although the influenza pandemic explains much of the recent interest in the basic reproduction number, this interest is not limited to the field of influenza. Web of Science searches on the terms “reproduction number” or “reproductive number” revealed that there have been 710 publications on this topic from 2009 through February 7, 2013, across various disciplines, with most articles being published in journals covering infectious diseases and mathematical modeling. Table A (available as a supplement to this article at http://www.ajph.org) shows breakdown by journal. If the search is expanded to include data from previous years, it is clear that there has been exponential growth by calendar year in the number of publications on this topic (Figure 1). Why is there such growing interest in R₀ among the disciplines interested in the dynamics of infectious diseases? To help better understand the interest in the basic reproduction number among public health officials, infectious disease researchers, and theoretical modelers, we reviewed the derivation of R₀ and its history.Open in a separate windowFIGURE 1—The number of publications regarding infectious disease and mathematical modeling as reported by Web of Science.Note. The figure was produced by searching Web of Science on the terms “reproduction number” or “reproductive number” and limiting the results to the fields of infectious diseases, mathematical computational biology, and applied mathematics. Clearly, interest in research regarding the basic reproductive number has risen dramatically since the 1990s. The number of publications in this area currently appears to be growing exponentially.We present a basic epidemiological compartmental model (a susceptible–infected–recovered or SIR model with S, I, and R representing the 3 compartments) described by Kermack and McKendrick.12 In this relatively simple model designed to describe epidemics, individuals start as susceptible to a particular pathogen and then progress to the other 2 compartments if infected. The model is defined by a system of 3 ordinary differential equations (ODEs):in which β is the transmission rate, γ is the recovery rate (or the inverse of the infectious period), and N is the total population size such that N = S + I + R. The standard model in equation 1 assumes no births or deaths. At the beginning of the outbreak or epidemic (t = 0) we assume the population is composed entirely of susceptible individuals and a single infectious individual. With this model, if the transmission rate exceeds the recovery rate (i.e., β/γ > 1), the disease will spread (dI/dt > 0). Alternatively, β/γ is the number of new infections per unit time multiplied by the time period of infectiousness, and describes the number of new infections resulting from the initially infected individual. In the presented case of the simple SIR model, the basic reproduction number (or ratio) equals β/γ.The scientific community largely underappreciated the implications of the Kermack–McKendrick model until the late 1970s, when Anderson and May13 used the model to study strategies for controlling infectious diseases. R₀ is a parameter of importance for gauging the disease dynamics because it indicates when an outbreak might happen based on the threshold value of 1.0. More generally, if the effective reproduction number R_e = R₀ × (S/N) is greater than 1.0, we predict that the disease continues its spread; the effective reproduction reflects the fact that, as proportion of susceptible individuals decreases (S/N), disease transmission slows. From this simple mathematical perspective, epidemiologists frequently consider the basic reproductive number one of the most vital parameters in determining whether an epidemic is “controllable.”14,15 The objective of any public health response during an influenza pandemic, for example, is to slow or stop the spread of the virus by employing mitigation strategies that either (1) reduce R₀ by changing the transmission rate (e.g., via school closure) or the duration of infectiousness (e.g., through antiviral use) or (2) reduce R_e by reducing the number of susceptible individuals (e.g., by vaccination).